Relevant bibliographies by topics / Triangle Inequality

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Triangle Inequality'

Author: Grafiati
Published: 4 June 2021
Last updated: 21 February 2023

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Journal articles on the topic "Triangle Inequality"

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Hidayatin, Nur, and Frida Murtinasari. "Generalisasi Ketaksamaan Sinus pada Segitiga." Jurnal Axioma : Jurnal Matematika dan Pembelajaran 7, no. 1 (June 6, 2022): 72–78. http://dx.doi.org/10.56013/axi.v7i1.1195.

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This study aims to find a generalization of the sine inequality of any triangles. This generalization is the general form of the sine inequality in a triangle when the angles given are not angles of the triangle, i.e. when the sum of the three angles is not equal to . The sine inequality that will be studied focuses on the inequalities of the sum and multiplication of sine in triangles. In the process, qualitative research methods are carried out in the form of literature review, namely studying the sum and the multiplication inequalities of sine in triangles which will then be developed and obtained new generalizations from the previous inequalities, namely generalizations of sine inequalities in triangles. These generalizations include generalizing the inequalities of the sum of sine in triangles and generalizing the inequalities of multiplication sine in triangles. To study this, it is necessary to first study the concepts of trigonometry, namely the definition of sine and cosine, the rules of sine and cosine; the relationship of the sine cosine to the sides of the triangle; the relationship of the radius of the circumcircle of the triangle to the sides and angles of the triangle; and arithmetic and geometric mean inequalities.The results of this study obtained the generalization of the sine inequality of any triangles. Keywords: inequality, sine, triangle
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Yang, Zheng, Lirong Jian, Chenshu Wu, and Yunhao Liu. "Beyond triangle inequality." ACM Transactions on Sensor Networks 9, no. 2 (March 2013): 1–20. http://dx.doi.org/10.1145/2422966.2422983.

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Shevchishin, Vsevolod, and Gleb Smirnov. "Symplectic triangle inequality." Proceedings of the American Mathematical Society 148, no. 4 (January 6, 2020): 1389–97. http://dx.doi.org/10.1090/proc/14842.

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Hoehn, Larry. "Geometrical Inequalities via Bisectors." Mathematics Teacher 82, no. 2 (February 1989): 96–99. http://dx.doi.org/10.5951/mt.82.2.0096.

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The proofs of several theorems in secondary school mathematics that involve geometrical inequalities are more complicated than they really need to be. This article presents an easier-to-understand alternative to the usual proofs of inequalities in triangles. The only inequality with which we need to assume familiarity is the triangle inequality (i.e., the sum of any two sides of a plane triangle is greater than the third side).
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Greenhoe, D. "Properties of distance spaces with power triangle inequalities." Carpathian Mathematical Publications 8, no. 1 (June 30, 2016): 51–82. http://dx.doi.org/10.15330/cmp.8.1.51-82.

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Metric spaces provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the triangle inequality. However, there are several applications in which the triangle inequality does not hold but in which we may still like to perform analysis. This paper investigates what happens if the triangle inequality is removed all together, leaving what is called a distance space, and also what happens if the triangle inequality is replaced with a much more general two parameter relation, which is herein called the "power triangle inequality". The power triangle inequality represents an uncountably large class of inequalities, and includes the triangle inequality, relaxed triangle inequality, and inframetric inequality as special cases. The power triangle inequality is defined in terms of a function that is herein called the power triangle function. The power triangle function is itself a power mean, and as such is continuous and monotone with respect to its exponential parameter, and also includes the operations of maximum, minimum, mean square, arithmetic mean, geometric mean, and harmonic mean as special cases.
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Bencze, Mihaly, and GCHQ Problems Group. "A Triangle Inequality: 10644." American Mathematical Monthly 106, no. 5 (May 1999): 476. http://dx.doi.org/10.2307/2589167.

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Bailey, Herbert R., and Robert Bannister. "A Stronger Triangle Inequality." College Mathematics Journal 28, no. 3 (May 1997): 182. http://dx.doi.org/10.2307/2687521.

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Shahbari, Juhaina A., and Moshe Stupel. "106.13 A triangle inequality." Mathematical Gazette 106, no. 565 (February 24, 2022): 138. http://dx.doi.org/10.1017/mag.2022.28.

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Bailey, Herbert R., and Robert Bannister. "A Stronger Triangle Inequality." College Mathematics Journal 28, no. 3 (May 1997): 182–86. http://dx.doi.org/10.1080/07468342.1997.11973859.

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Retkes, Zoltan. "Generalising a triangle inequality." Mathematical Gazette 102, no. 555 (October 17, 2018): 422–27. http://dx.doi.org/10.1017/mag.2018.108.

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The main goal of this paper is to give a deeper understanding of the geometrical inequality proposed by Martin Lukarevski in [1]. In order to formulate our results we shall introduce and use the following notation throughout this paper. Let A1A2A3 be a triangle a1, a2, a3, the lengths of the sides opposite to A1, A2, A3 respectively, P an arbitrary inner point of it xi, the distance of P from the side of length ai. Let r, R be the inradius and circumradius of the triangle hi, the altitude belonging to side ai, Δ the area and finally let α be a real parameter. We adopt also the use of Σui to refer the sum taken over the suffices i = 1, 2, 3. Now we are in the position to reformulate the original problem into a more general form namely: find bounds for Σxαi in terms of r and R. The main results of our investigation are summarised in the following theorem.
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Dissertations / Theses on the topic "Triangle Inequality"

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Narreddy, Naga Sambu Reddy, and Tuğrul Durgun. "Clusters (k) Identification without Triangle Inequality : A newly modelled theory." Thesis, Uppsala universitet, Institutionen för informatik och media, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-183608.

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Cluster analysis characterizes data that are similar enough and useful into meaningful groups (clusters).For example, cluster analysis can be applicable to find group of genes and proteins that are similar, to retrieve information from World Wide Web, and to identify locations that are prone to earthquakes. So the study of clustering has become very important in several fields, which includes psychology and other social sciences, biology, statistics, pattern recognition, information retrieval, machine learning and data mining [1] [2].   Cluster analysis is the one of the widely used technique in the area of data mining. According to complexity and amount of data in a system, we can use variety of cluster analysis algorithms. K-means clustering is one of the most popular and widely used among the ten algorithms in data mining [3]. Like other clustering algorithms, it is not the silver bullet. K-means clustering requires pre analysis and knowledge before the number of clusters and their centroids are determined. Recent studies show a new approach for K-means clustering which does not require any pre knowledge for determining the number of clusters [4].   In this thesis, we propose a new clustering procedure to solve the central problem of identifying the number of clusters (k) by imitating the desired number of clusters with proper properties. The proposed algorithm is validated by investigating different characteristics of the analyzed data with modified theory, analyze parameters efficiency and their relationships. The parameters in this theory include the selection of embryo-size (m), significance level (α), distributions (d), and training set (n), in the identification of clusters (k).
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Berman, Andrew P. "Efficient content-based retrieval of images using triangle-inequality-based algorithms /." Thesis, Connect to this title online; UW restricted, 1999. http://hdl.handle.net/1773/6989.

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Hamilton, Jeremy. "An Exploration of the Erdös-Mordell Inequality." Youngstown State University / OhioLINK, 2010. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1287605197.

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OTSUBO, Shigeru, and Yumeka HIRANO. "Poverty-Growth-Inequality Triangle under Globalization: Time Dimensions and the Control Factors of the Impacts of Integration." 名古屋大学大学院国際開発研究科, 2012. http://hdl.handle.net/2237/16949.

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Wigren, Thomas. "The Cauchy-Schwarz inequality : Proofs and applications in various spaces." Thesis, Karlstads universitet, Avdelningen för matematik, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:kau:diva-38196.

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We give some background information about the Cauchy-Schwarz inequality including its history. We then continue by providing a number of proofs for the inequality in its classical form using various proof techniques, including proofs without words. Next we build up the theory of inner product spaces from metric and normed spaces and show applications of the Cauchy-Schwarz inequality in each content, including the triangle inequality, Minkowski's inequality and Hölder's inequality. In the final part we present a few problems with solutions, some proved by the author and some by others.
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Johnson, Timothy Kevin. "A reformulation of Coombs' Theory of Unidimensional Unfolding by representing attitudes as intervals." Thesis, The University of Sydney, 2004. http://hdl.handle.net/2123/612.

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An examination of the logical relationships between attitude statements suggests that attitudes can be ordered according to favourability, and can also stand in relationships of implication to one another. The traditional representation of attitudes, as points on a single dimension, is inadequate for representing both these relations but representing attitudes as intervals on a single dimension can incorporate both favourability and implication. An interval can be parameterised using its two endpoints or alternatively by its midpoint and latitude. Using this latter representation, the midpoint can be understood as the �favourability� of the attitude, while the latitude can be understood as its �generality�. It is argued that the generality of an attitude statement is akin to its latitude of acceptance, since a greater semantic range increases the likelihood of agreement. When Coombs� Theory of Unidimensional Unfolding is reformulated using the interval representation, the key question is how to measure the distance between two intervals on the dimension. There are innumerable ways to answer this question, but the present study restricts attention to eighteen possible �distance� measures. These measures are based on nine basic distances between intervals on a dimension, as well as two families of models, the Minkowski r-metric and the Generalised Hyperbolic Cosine Model (GHCM). Not all of these measures are distances in the strict sense as some of them fail to satisfy all the metric axioms. To distinguish between these eighteen �distance� measures two empirical tests, the triangle inequality test, and the aligned stimuli test, were developed and tested using two sets of attitude statements. The subject matter of the sets of statements differed but the underlying structure was the same. It is argued that this structure can be known a priori using the logical relationships between the statement�s predicates, and empirical tests confirm the underlying structure and the unidimensionality of the statements used in this study. Consequently, predictions of preference could be ascertained from each model and either confirmed or falsified by subjects� judgements. The results indicated that the triangle inequality failed in both stimulus sets. This suggests that the judgement space is not metric, contradicting a common assumption of attitude measurement. This result also falsified eleven of the eighteen �distance� measures because they predicted the satisfaction of the triangle inequality. The aligned stimuli test used stimuli that were aligned at the endpoint nearest to the ideal interval. The results indicated that subjects preferred the narrower of the two stimuli, contrary to the predictions of six of the measures. Since these six measures all passed the triangle inequality test, only one measure, the GHCM (item), satisfied both tests. However, the GHCM (item) only passes the aligned stimuli tests with additional constraints on its operational function. If it incorporates a strictly log-convex function, such as cosh, the GHCM (item) makes predictions that are satisfied in both tests. This is also evidence that the latitude of acceptance is an item rather than a subject or combined parameter.
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Johnson, Timothy Kevin. "A reformulation of Coombs' Theory of Unidimensional Unfolding by representing attitudes as intervals." University of Sydney. Psychology, 2004. http://hdl.handle.net/2123/612.

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An examination of the logical relationships between attitude statements suggests that attitudes can be ordered according to favourability, and can also stand in relationships of implication to one another. The traditional representation of attitudes, as points on a single dimension, is inadequate for representing both these relations but representing attitudes as intervals on a single dimension can incorporate both favourability and implication. An interval can be parameterised using its two endpoints or alternatively by its midpoint and latitude. Using this latter representation, the midpoint can be understood as the �favourability� of the attitude, while the latitude can be understood as its �generality�. It is argued that the generality of an attitude statement is akin to its latitude of acceptance, since a greater semantic range increases the likelihood of agreement. When Coombs� Theory of Unidimensional Unfolding is reformulated using the interval representation, the key question is how to measure the distance between two intervals on the dimension. There are innumerable ways to answer this question, but the present study restricts attention to eighteen possible �distance� measures. These measures are based on nine basic distances between intervals on a dimension, as well as two families of models, the Minkowski r-metric and the Generalised Hyperbolic Cosine Model (GHCM). Not all of these measures are distances in the strict sense as some of them fail to satisfy all the metric axioms. To distinguish between these eighteen �distance� measures two empirical tests, the triangle inequality test, and the aligned stimuli test, were developed and tested using two sets of attitude statements. The subject matter of the sets of statements differed but the underlying structure was the same. It is argued that this structure can be known a priori using the logical relationships between the statement�s predicates, and empirical tests confirm the underlying structure and the unidimensionality of the statements used in this study. Consequently, predictions of preference could be ascertained from each model and either confirmed or falsified by subjects� judgements. The results indicated that the triangle inequality failed in both stimulus sets. This suggests that the judgement space is not metric, contradicting a common assumption of attitude measurement. This result also falsified eleven of the eighteen �distance� measures because they predicted the satisfaction of the triangle inequality. The aligned stimuli test used stimuli that were aligned at the endpoint nearest to the ideal interval. The results indicated that subjects preferred the narrower of the two stimuli, contrary to the predictions of six of the measures. Since these six measures all passed the triangle inequality test, only one measure, the GHCM (item), satisfied both tests. However, the GHCM (item) only passes the aligned stimuli tests with additional constraints on its operational function. If it incorporates a strictly log-convex function, such as cosh, the GHCM (item) makes predictions that are satisfied in both tests. This is also evidence that the latitude of acceptance is an item rather than a subject or combined parameter.
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8

Hsu, Chih-Yung, and 許至勇. "Refinements of triangle inequality and Jensen’s inequality." Thesis, 2007. http://ndltd.ncl.edu.tw/handle/86930617569692611296.

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碩士
國立中央大學
數學研究所
95
In this thesis, we prove a sharp triangle inequality and its reverse inequality for strongly integrable functions with values in a Banach space X. This contains as a special case a recent result of Kato et al on sharp triangle inequality for n elements. We also discuss a generalized triangle inequality for Lp functions with values in X. It contains as a special case the triangle inequality of the second kind for two elements, which is implied by the Euler-Lagrange type identity. Besides, some properties related to a refined Jensen’s inequality are observed.
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Wang, Huan-Yu, and 王煥宇. "A Fast Similarity Algorithm for Personal Ontologies Using Triangle Inequality." Thesis, 2009. http://ndltd.ncl.edu.tw/handle/93888608461842522310.

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碩士
國立中興大學
資訊科學與工程學系所
97
The Personal Ontology Recommender System (PORE) currently operated in the library of National Chung Hsing University is a recommender system developed by our research team. The system consists of content-based recommendation model based on personal ontology and collaborative filtering recommendation model. For collaborative filtering, the recommender system needs to compute the similarity between any two users. That will incur lots of computations because the library currently has more than thirty thousands of users and three hundred thousands of collections. The purpose of this thesis is to design an efficient algorithm for computing the similarity between two users. A personal ontology representing the favorites of a user in PORE is a tree structure. In this thesis, we define tree distance for measuring the dissimilarity between two users. We then propose an efficient algorithm for calculating ontology similarities using triangle inequality. The experimental results show that the proposed method can save up to 88% of comparisons compared to that of brute force algorithm.
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CHING, HOU HSUEH, and 侯雪卿. "The Case Study of learning triangle inequality of Elementary School Students." Thesis, 2019. http://ndltd.ncl.edu.tw/handle/fqa7c6.

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博士
國立中正大學
教育學研究所
107
The purpose of this study is to investigate grade 4 students’ learning performance and learning outcomes in the topic of triangle inequality. This study uses a case study approach. Three grade 4 students in different achievement level attended in a sequence of learning activities in the investigative approach. The learning performance and learning outcomes are analyzed and interpreted by classroom observing, interviews, work sheets, pre- and post- assessment, and self-reflection of the instructor. The results show that:1.Grade 4 students are able to understand the triangle inequality through the phases of image constructing, property noticing, and formalizing. They are able to find that three line segments do not necessarily form a triangle, existence condition of a triangle on the sides, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side,and to find the length of third side in a triangle with two known sides. However, the learning path of students in different achievement level are different. 2. Students of high and medium achievement level are able to solve all kinds of triangle inequality problems successfully in the posttest. The low achievement level student are able to solve problems of existence condition of a triangle on the sides and finding the length of third side in a triangle with two known sides. However, there is misconcepiton about three line segments do not necessarily form a triangle and the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
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Books on the topic "Triangle Inequality"

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Socolovsky, Eduardo A. A dissimilarity measure for clustering high- and infinite dimensional data that satisfies the triangle inequality. Hampton, VA: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 2002.

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Book chapters on the topic "Triangle Inequality"

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Denny, William Gozali, and Ruli Manurung. "SOM Training Optimization Using Triangle Inequality." In Advances in Self-Organizing Maps and Learning Vector Quantization, 61–71. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-28518-4_5.

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Schubert, Erich. "A Triangle Inequality for Cosine Similarity." In Similarity Search and Applications, 32–44. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-89657-7_3.

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Chandran, L. Sunil, and L. Shankar Ram. "Approximations for ATSP with Parametrized Triangle Inequality." In STACS 2002, 227–37. Berlin, Heidelberg: Springer Berlin Heidelberg, 2002. http://dx.doi.org/10.1007/3-540-45841-7_18.

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Manyem, Prabhu. "Constrained spanning, Steiner trees and the triangle inequality." In Springer Optimization and Its Applications, 355–67. New York, NY: Springer New York, 2009. http://dx.doi.org/10.1007/978-0-387-98096-6_19.

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Kowalik, Łukasz, and Marcin Mucha. "Two Approximation Algorithms for ATSP with Strengthened Triangle Inequality." In Lecture Notes in Computer Science, 471–82. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-03367-4_41.

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Böckenhauer, Hans-Joachim, Karin Freiermuth, Juraj Hromkovič, Tobias Mömke, Andreas Sprock, and Björn Steffen. "The Steiner Tree Reoptimization Problem with Sharpened Triangle Inequality." In Lecture Notes in Computer Science, 180–91. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-13073-1_17.

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Lumezanu, Cristian, Randy Baden, Neil Spring, and Bobby Bhattacharjee. "Triangle Inequality and Routing Policy Violations in the Internet." In Lecture Notes in Computer Science, 45–54. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-00975-4_5.

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Böckenhauer, Hans-Joachim, Dirk Bongartz, Juraj Hromkovič, Ralf Klasing, Guido Proietti, Sebastian Seibert, and Walter Unger. "On k-Edge-Connectivity Problems with Sharpened Triangle Inequality." In Lecture Notes in Computer Science, 189–200. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003. http://dx.doi.org/10.1007/3-540-44849-7_24.

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Kryszkiewicz, Marzena, and Piotr Lasek. "A Neighborhood-Based Clustering by Means of the Triangle Inequality." In Intelligent Data Engineering and Automated Learning – IDEAL 2010, 284–91. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-15381-5_35.

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Bender, Michael A., and Chandra Chekuri. "Performance Guarantees for the TSP with a Parameterized Triangle Inequality." In Lecture Notes in Computer Science, 80–85. Berlin, Heidelberg: Springer Berlin Heidelberg, 1999. http://dx.doi.org/10.1007/3-540-48447-7_10.

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Conference papers on the topic "Triangle Inequality"

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Berman, Andrew P., and Linda G. Shapiro. "Triangle-inequality-based pruning algorithms with triangle tries." In Electronic Imaging '99, edited by Minerva M. Yeung, Boon-Lock Yeo, and Charles A. Bouman. SPIE, 1998. http://dx.doi.org/10.1117/12.333855.

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Acharyya, Sreangsu, Arindam Banerjee, and Daniel Boley. "Bregman Divergences and Triangle Inequality." In Proceedings of the 2013 SIAM International Conference on Data Mining. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2013. http://dx.doi.org/10.1137/1.9781611972832.53.

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Lumezanu, Cristian, Randy Baden, Neil Spring, and Bobby Bhattacharjee. "Triangle inequality variations in the internet." In the 9th ACM SIGCOMM conference. New York, New York, USA: ACM Press, 2009. http://dx.doi.org/10.1145/1644893.1644914.

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Wang, Guohui, Bo Zhang, and T. S. Eugene Ng. "Towards network triangle inequality violation aware distributed systems." In the 7th ACM SIGCOMM conference. New York, New York, USA: ACM Press, 2007. http://dx.doi.org/10.1145/1298306.1298331.

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Kaafar, M. A., F. Cantin, B. Gueye, and G. Leduc. "Detecting Triangle Inequality Violations for Internet Coordinate Systems." In 2009 IEEE International Conference on Communications Workshops. IEEE, 2009. http://dx.doi.org/10.1109/iccw.2009.5207998.

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Ghamdi, Sami Al, and Giuseppe Di Fatta. "Efficient Parallel K-Means on MapReduce Using Triangle Inequality." In 2017 IEEE 15th Intl Conf on Dependable, Autonomic and Secure Computing, 15th Intl Conf on Pervasive Intelligence and Computing, 3rd Intl Conf on Big Data Intelligence and Computing and Cyber Science and Technology Congress(DASC/PiCom/DataCom/CyberSciTech). IEEE, 2017. http://dx.doi.org/10.1109/dasc-picom-datacom-cyberscitec.2017.163.

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He, Chunxia, Jinyi Chang, and Xiaoyun Chen. "Using the Triangle Inequality to Accelerate TTSAS Cluster Algorithm." In 2010 International Conference on Electrical and Control Engineering (ICECE 2010). IEEE, 2010. http://dx.doi.org/10.1109/icece.2010.620.

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Gentile, Camillo. "Distributed Sensor Location through Linear Programming with Triangle Inequality Constraints." In 2006 IEEE International Conference on Communications. IEEE, 2006. http://dx.doi.org/10.1109/icc.2006.255710.

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Zhang, Tongquan, and Ying Yin. "Travelling Production Line Problem on Digraphs with Parameterized Triangle Inequality." In 2010 International Conference on Computational Intelligence and Software Engineering (CiSE). IEEE, 2010. http://dx.doi.org/10.1109/cise.2010.5676930.

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Biswas, Arijit, and David Jacobs. "An Efficient Algorithm for Learning Distances that Obey the Triangle Inequality." In British Machine Vision Conference 2015. British Machine Vision Association, 2015. http://dx.doi.org/10.5244/c.29.10.

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